We covered section 4.4.
If B = {b1..bn} is a basis of R^n and x is in R^n then

	x = P_B [x]_B

which means that P_B sends the coordinate-vector-of-x to x.
That also means that (P_B)^(-1) sends x to its coordinate-vector.

We also covered section 4.5 but this was mostly material we have
seen before.

In general, these three things are not equivalent:
	* lin. indep.
	* basis
	* spanning set
However, if S is a set with dim(V) elements of V,  and if dim(V) < infinity
then the above three statements become equivalent. That can help you
to take short-cuts.

In section 4.6 we learned that if you row-reduce a matrix A then
the following things stay the same:
	rank
	dim(ColumnSpace)   (that's equal to the rank)
	RowSpace

So under row-reduction, the RowSpace stays the same, the ColumnSpace
can change but its dimension stays the same. Moreover, both
the RowSpace and the ColumnSpace have the same dimension, their
dimension equals the rank.

In Section 4.7 we discussed the change-of-basis matrix. I put one
exercise on the board.

Re-read all these sections.

Turn in questions:

Section 4.4.  Ex 10, 14, 28.  Read through the questions, especially
the True/False questions (ask about them if you see one that's not clear).

Section 4.5.  Ex 4, 8, 14. Again, read the section and the questions
including the True/False questions.

Section 4.6.  Ex 2 (no computation, RREF is already given).
Observe that Ex 5-16 are all variations on Theorem 14.
Read the True/False questions (Ex 17,...) to test your understanding.

Section 4.7.  Ex 2.

One more comment on section 4.7:  at the end of page 243 you see a formula
to compute the change-of-basis matrix. But note that this formula is only
for bases of R^n.  For any vector space other than R^n you'll need to use
formula (5) in Theorem 15.